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"In mathematics, we all have come across problem statements like the height of all the students of the class is less than 168 cm, the classroom can hold at most 90 tables/chairs or both, etc. Here we get some statements comprising a sign ‘&lt;’ (less than), ‘&gt;’ (greater than), ‘≤’ (less than or equal), '≥' (greater than or equal), and '≠'(not equal to) which are acknowledged as inequalities. Learn about Linear Inequations in this article.\r\n\r\nThis implies when two real numbers or algebraic expressions are associated by the symbol ‘&lt;’, ‘&gt;’, ‘≤’ and ‘≥’ forms an inequality.\r\n\r\n \tInequality statement with ‘&lt;’, and ‘&gt;’ signs forms strict inequalities.\r\n \tInequality statement with ‘≤’ and ‘≥’ signs forms slack inequalities.\r\n \tAn inequality statement with one variable is termed as linear inequalities in one variable.\r\n \tAn inequality statement with two variables is termed as linear inequalities in two variables.\r\n \tSo, when two linear algebraic expressions in one variable are related by the symbol ‘&lt;’, ‘&gt;’, ‘≤’ and ‘≥’ forms a linear inequality in one variable.\r\n \tFor example: 3x + 5 &lt; 10.\r\n \tSimilarly, when two linear algebraic expressions in two variables are related by the symbol ‘&lt;’, ‘&gt;’, ‘≤’ and ‘≥’ forms a linear inequality in two variables.\r\n \tFor example: 3x + 5y ≥ 20.\r\n\r\nLearn about Linear Inequalities in this video\r\n\r\n&lt;iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/HBRnW1ibMw4\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen?rel=0&gt;&lt;/iframe&gt;\r\nLinear Inequalities In One Variable\r\nWhen two linear algebraic expressions in one variable are related by the symbol ‘&lt;’, ‘&gt;’, ‘≤’ and ‘≥’ forms a linear inequality in one variable. For example: 9x + 5 &lt; 15.\r\n\r\nIn this section, we will learn about how to solve a linear inequation in one variable. Some of the rules which we need to keep in mind while solving a linear inequation which are:\r\nRule #1: Addition or Subtraction\r\nWhen equal numbers are added (or subtracted) on (from) both the sides of inequalities then the sign of inequality does not change. consider the below example to understand the concept.\r\nFor example\r\nLet 3x + 5 &lt; 10 be the given inequality.\r\n\r\nNow by subtracting 5 from both the sides of the inequality, we get:\r\n\r\n3x + 5 – 5 &lt; 10 – 5\r\n\r\n⇒ 3x &lt; 5\r\nRule #2: Multiplication or Division\r\nWhen both sides of the inequality are multiplied or divided by the same positive number then the sign of inequality does not change. Whereas, if both the sides are multiplied or divided by the same negative number then the sign of inequality gets reversed.\r\nFor example\r\nLet 3x &lt; 5 be the given inequality.\r\n\r\nNow, when both the sides of the inequality are divided by 3, we get:\r\n\r\nx &lt; 5 / 3.\r\n\r\nWhereas if both the sides of the given inequality are divided by – 3, we get:\r\n\r\n- x &gt; - 5 / 3.\r\nGraphical Representation of Solution of a Linear Inequality in One Variable.\r\nLet’s understand the graphical representation of a linear inequality in one variable with the help of an example.\r\n\r\nSolve: 5x – 3 &lt; 3x + 1, where x ∈ R \r\n\r\nGiven: 5x – 3 &lt; 3x + 1\r\n\r\nNow by subtracting 3x from both the sides of the given inequality, we get;\r\n\r\n⇒ (5x - 3) – 3x &lt; (3x + 1) – 3x\r\n\r\n⇒ 2x – 3 &lt; 1\r\n\r\nNow by adding 3 on both the sides of the above inequality we get;\r\n\r\n⇒ 2x &lt; 4\r\n\r\nNow by dividing both the sides of the above inequality by 2, we get;\r\n\r\n⇒ x &lt; 2\r\n\r\n∵ x ∈ R ⇒ x = (- ∞, 2)\r\n\r\nThe graphical representation of the solution of the inequality 5x – 3 &lt; 3x + 1, where x ∈ R i\r\n\r\nTherefore, any solution of an inequality in one variable is a value of the variable which makes it a true declaration. This indicates we have found the solutions to the inequality by trial and error method.\r\n\r\n\r\nLinear Inequalities In Two Variables\r\nWhen two linear algebraic expressions in two variablesare related by the symbol ‘&lt;’, ‘&gt;’, ‘≤’ and ‘≥’ forms a linear inequality in two variables.\r\n\r\nFor example: 9x + 4y ≥ 20.\r\n\r\nIn this section, we will learn how to solve a linear inequation with the help of an example:\r\n\r\nSolve: 2x + y &gt; 6 \r\n\r\nGiven: 2x + y &gt; 6\r\n\r\nStep 1:Replace the inequality present in the given inequation with “=” we get 2x + y = 6.\r\n\r\nStep -2: Represent the equation 2x + y = 6 on a 2D plane as shown below:\r\n\r\n\r\n\r\nStep 3: Now, put x = 0 and y = 0 in the given inequality: 2x + y &gt; 6 we get\r\n\r\n0 + 0 &gt; 6, and this is false.\r\n\r\n\r\n\r\nHence, shade the portion which does not contain the point (0, 0) as shown in the figure above. The shaded portion in the above-given diagram represents the solution of the given inequality.\r\nSolution Of The System Of Linear Inequation In Two Variables\r\nIn this section we will learn about how to solve a system of linear inequation in two variables with the help of an example:\r\n\r\nSolve the system of inequation: 5x + 4y ≤ 40, x ≥ 2 and y ≥ 3 \r\n\r\nGiven: 5x + 4y ≤ 40, x ≥ 2 and y ≥ 3\r\n\r\nStep 1: Plot all the given inequation on a 2D plane as shown in the previous section.\r\n\r\nStep 2: Shade the portion which is common to all the given inequation.\r\n\r\nAs shown in the figure below:\r\n\r\n\r\nKey Terms\r\n\r\n \tThe region containing all the solutions of an inequality is called the solution region.\r\n \tTo recognize the plane represented by inequality, it is just sufficient to take any point (a,b) (not on the line) and examine whether it satisfies the inequality or not.\r\n \tIf it satisfies, then the inequality represents the plane and shade the region which contains the point, otherwise, the inequality signifies that plane that does not include the point within it.\r\n \tFor convenience, the point (0, 0) is preferred.\r\n \tIf inequality is of the type ax + by ≥ c or ax + by ≤ c, then the points on the line ax + by = c; are also incorporated in the solution region. So form a dark line in the solution region.\r\n \tIf inequality is of the form ax + by &gt; c or ax + by &lt; c, then the points on the line ax + by = c; are not to be incorporated in the solution region. So form a broken or dotted line in the solution region.\r\n\r\n\r\n\r\n\r\n\r\nIf you are checking Linear Inequality article, also check the related maths articles in the table below:\r\n\r\n\r\nMethods of Differentiation\r\nMultiple Line Graph\r\n\r\n\r\nLimits And Continuity\r\nLogarithmic Functions\r\n\r\n\r\nPolynomial Identities Algebra\r\nApplication of Integrals\r\n\r\n\r\n\r\n\r\nStay tuned to the Testbook app or visit the Testbook website for more updates on such similar topics from mathematics, science, and numerous such subjects, and can even check the test series available to test your knowledge regarding various exams.\r\nLinear Inequality FAQs\r\n[faq_accordion]\r\n\r\n[mks_accordion_item title=\"What are various signs used to represent inequality?\" number=1]The signs used are as follows; ‘&lt;’ (less than), ‘&gt;’ (greater than), ‘≤’ (less than or equal), '≥' (greater than or equal), and '≠'(not equal to).\r\n\r\n[/mks_accordion_item]\r\n\r\n[mks_accordion_item title=\"What is one variable linear inequality?\" number=2]An inequality statement with one variable is termed as linear inequalities in one variable.\r\nFor example: 7x + 8&lt; 10.\r\n\r\n[/mks_accordion_item]\r\n\r\n[mks_accordion_item title=\"What is two-variable linear inequality?\" number=3]An inequality statement with two variables is termed as linear inequalities in two variables.\r\nFor example: 9x + 4y ≥ 50.\r\n\r\n[/mks_accordion_item]\r\n\r\n[mks_accordion_item title=\"What are strict inequalities?\" number=4]Inequality statement with ‘&lt;’, and ‘&gt;’ signs forms strict inequalities.\r\n\r\n[/mks_accordion_item]\r\n\r\n[mks_accordion_item title=\"What are slack inequalities?\" number=5]Inequality statement with ‘≤’ and ‘≥’ signs forms slack inequalities.\r\n\r\n[/mks_accordion_item]\r\n\r\n[/faq_accordion]"
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